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In representation theory, the notion of twisted intertwiners is a generalization of that of intertwiners.

References using the exact terminology “twisted intertwiner” include Fuchs & Schweigert 2000a (7.2), 2000b (2.2), Felder, Fröhlich, Fuchs & Schweigert 2000 (4.7), but the notion itself is older and may be known under other names.

Definition

Definition

For $G$ a group and $\rho_1, \rho_2 \;\in\; Rep(G)$ a pair of linear representation on vector spaces $V_1$, $V_2$, respectively (all this generalizes straightforwardly to algebras and their modules), a twisted intertwiner

$$\rho_1 \Rightarrow \rho_2$$

is

1. an automorphism $\alpha \in Aut(G)$ of the group

2. a linear map $\eta \colon V_1 \longrightarrow V_2$

such that for all $g \in G$ we have

$$\eta \circ \rho_1(g) \;=\; \rho_2\big(\alpha(g)\big) \circ \eta \,.$$

Given a pair of twisted intertwiners $\rho_1 \overset{(\alpha,\eta)}{\Rightarrow} \rho_2 \overst{(\alpha',\eta')}{\Rightarrow} \rho_3$ their composition is given by composing their components separately.

Properties

Category-theoretically

Writing $\mathbf{B}G$ for the delooping groupoid of $G$ and $Vec$ for the category of vector spaces (over a given ground field), the ordinary category Rep of $G$-representations and ordinary intertwiners $\eta \colon \rho_1 \Rightarrow \rho_2$ between these is equivalently the functor category

$$Rep(G) \;\;\simeq\;\; Func\big( \mathbf{B}G ,\, Vec \big) \,.$$

In contrast, the category of $G$-representations with twisted intertwiners between them is the iso-comma (2,1)-category between $\mathbf{B}G \colon \mathbf{B}Aut(G) \longrightarrow Grpd$ and $Vec \colon \ast \to Grpd$, whose 1-morphisms are diagrams in Grpd of this form:

and whose 2-morphisms are the evident paper-cup pasting diagrams

References

• Giovanni Felder, Jürg Fröhlich, Jürgen Fuchs, Christoph Schweigert: The geometry of WZW branes, J. Geom. Phys. 34 (2000) 162-190 [doi:10.1016/S0393-0440(99)00061-3, arXiv:hep-th/9909030]

• Jürgen Fuchs, Christoph Schweigert: Symmetry breaking boundaries II. More structures; examples, Nucl. Phys. B 568 (2000) 543-593 [arXiv:hep-th/9908025, doi:10.1016/S0550-3213(99)00669-0]

• Jürgen Fuchs, Christoph Schweigert: Lie algebra automorphisms in conformal field theory, in Conference on Infinite Dimensional Lie Theory and Conformal Field Theory (May 2000) [arXiv:math/0011160]

Particle-hole symmetry.

Original

• S. M. Girvin: Particle-hole symmetry in the anomalous quantum Hall effect, Phys. Rev. B 29 (1984) 6012(R) [doi:10.1103/PhysRevB.29.6012]

$\neq \gt 1$ may be understood as $\nu = 1$ for majority spin polarization with $\nu - 1$ for the minority polarization (Haldane 1983 doi:10.1103/PhysRevLett.51.605)

Here

• “$\bullet$” means that the corresponding combination of $(k,q)$ is not admissible, in that $k q$ is odd or the Gauss sum $\sum_{n=0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} q n^2 } = 0$.

• “$\circ$” means that $\tfrac{q}{k} \gt 2$

So:

1. in the column of $q = 1$ all odd $k$ are excluded and all even $k$ are admissible,

2. in the column of $q = 2$ all odd $k$ are excluded and exactly every second even $k$ is admissible,

3. in rows of odd $k$ the odd $q$ are excluded, and for even $q = 2 r$ the result is admissible iff the Jacobi symbol $(r \vert k) \neq 0$.

   Since the Jacobi symbol $(r \vert k)$ vanishes iff $gcd(r,k) \neq 1$, this means that, at least for odd $k$, exactly only the reduced fractions appear.